Here we present a simple model for the evolution of an immersed, inextensible curve which incorporates linear viscoelastic effects of the surrounding fluid in a closed-form system of equations along the curve only.
The model is derived from the linear viscoelastic resistive force theory described in [1]}, [2]}, but requires some additional interpretation to yield a well-posed curve evolution. The resulting fiber evolution equations look like classical resistive force theory plus Euler beam theory [3]}, [4]}, [5]}, [6]}, but incorporate the evolution of an additional variable corresponding to a memory of the fiber curvature at previous times. This system of PDEs satisfies a very natural energy identity: in the absence of forcing, the bending energy of the filament plus an energy corresponding to the memory term are non-increasing in time. We prescribe a time-periodic forcing along the filament in the form of a preferred curvature and consider swimming as an emergent property. We show that this simple model, which is not coupled to any equations in the bulk, can capture the complexity of viscoelastic effects on swimming, including slowdowns and speedups relative to a Newtonian swimmer, depending on the form of the preferred curvature and the size of two viscoelastic parameters.
